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NONLINEAR OPTICAL
Annu. Rev. Mater. Sci. 1974.4:147-190. Downloaded from www.annualreviews.org
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PHENOMENA AND MATERIALS
Robert L. Byer
Department of Applied Physics, Stanford University, Stanford, California
94305
INTRODUCTION
The field of nonlinear optics has developed rapidly since its beginning in 1961. This
development is in both the theory of nonlinear effects and the theory of nonlinear
interactions in solids, and in the applications of nonlinear devices. This review
discusses nonlinear intcractions in solids and thc rcsultant nonlinear coupling of
electromagnetic waves that leads to second harmonic generation, optical mixing, and
optical parametric oscillation. Material requirements for device applications are
considered, and important nonlinear material properties summarized. At the outset,
a brief review of the development of nonlinear optics and devices is in order to provide
perspective of this rapidly growing field.
Historica l Review
In
1961, shortly after the demonstration of the laser, Franken et al (1) generated the
second harmonic of a Ruby laser in crystal quartz. The success of this experiment
relied directly on the enormous increase of power spectral brightness provided by
a laser source compared to incoherent sources. Power densities greater than
109 W/cm2 became available; these correspond to an electric field strength of
106 V cm 1. This field strength is comparable to atomic field strengths and, there­
-
fore, it was not too surprising that materials responded in a nonlinear manner to
the applied fields.
The early work in nonlincar optics concentrated on second harmonic generation.
Harmonic generation in the optical region is similar to the more familiar harmonic
generation at radio frequencies, with one important exception. In the radio frequency
range the wavelength is usually much larger than the harmonic generator, so that
the interaction is localized in a volume much smaller than the dimensions of a
wavelength. In the optical region the situation is usually reversed and the nonlinear
medium extends over many wavelengths. This leads to the consideration of propa­
gation effects since the electromagnetic wave interacts over an extended distance
with the generated nonlinear polarization. The situation is similar to a propagating
wave interacting with a phased linear dipole array. If this interaction is to be efficient,
147
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148
BYER
the phase of the propagating wave and the generated polarization must be proper. In
nonlinear optics this is referred to as phasematching. For second harmonic genera­
tion, phasematching implies that the phase velocity of the fundamental and second
harmonic waves are equal in the nonlinear material. Since optical materials are
dispersive, it is not possible to achieve equal phase velocities in isotropic materials.
Shortly after Franken et aI's first relatively inefficient non ph asematched second
harmonic generation experiment, Kleinman (2), Giordmaine (3), and Maker et al
(4), and later Akhmanov et al (5) showed that phase velocity matching could be
achieved in birefringent crystals by using the crystal birefringence to offset the
dispersion.
Along with the important concept of phasematching, other effects leading to
efficient second harmonic generation were studied. These included focusing (6, 7),
double refraction (8-10), and operation of second harmonic generators with an
external resonator (1 1 , 12) and within a laser cavity (13, 14).
An important extension of nonlinear interactions occurred in 1965 when Wang
& Racette ( 1 5) observed significant gain in a three-frequency mixing experiment. The
possibility of optical parametric gain had been previously considered theoretically
by Kingston (16), Kroll (17), Akhmanov & Khokhlov (18), and Armstrong et al ( 19).
It remained for Giordmaine & Miller (20) in 1 965 to achieve adequate parametric
gain in LiNb03 to overcome l osses and reach threshold for coherent oscillation.
This early work led to considerable activity in the study of parametric oscillators
as tunable coherent light sources.
Simultaneously with the activity in nonlinear devices, the theory of nonlinear
interactions received increased attention. It was recognized quite early that progress
in the field depended critically upon the availability of quality nonlinear materials.
Initially, the number of phasematchable nonlinear crystals with accurately measured
nonlinear coefficients was limited to a handful of previously known piezoelectric,
ferroelectric, or electro-optic materials. An important step in the problem of
searching for new nonlinear materials was made: when Miller (21) recognized that
the nonlinear susceptibi lity was related to the third power of the linear susceptibility
by a factor now known as Miller's delta. Whereas nonlinear coefficients of materials
span over four orders of magnitude, Miller's delta is constant to within 50%. To
the crystal grower and nonlinear materials sci(;ntist, this simple rule allows the
prediction of nonlinear coefficients based on known crystal indices of refraction and
symmetry, without having to carry out the expensive and time-consuming tasks of
crystal growth, accurate measurement of the birefringence to predict phasematching,
orientation, and finally second harmonic generation
The early progress in nonlinear optics has been the subject of a number of
monographs [Akhmanov & Khokhlov (22), Bloembergen (23), Butcher (24), Franken
& Ward (25)] and review articles [Ovander (26), Bonch-Bruevich & Khodovoi
(27), Minck et al (28), Pershan (28a), Akhmanov et al (29), Terhune & Maker (30),
Akhmanov & Khokhlov (31), and Kielich (32)]. In addition, nonlinear materials have
been reviewed by Suvorov & Sonin (33), Re:z (34), and Hulme (34a), and a
compilation of nonlinear materials is provided by Singh (35). Two books have
appeared, one a briefintroduction by Baldwin (36), and the second a clearly written
.
NONLINEAR OPTICAL PHENOMENA AND MATERIALS
text by Zernike & Midwinter
(37).
149
Finally, a text covering all aspects of nonlinear
optics is to appear soon (38).
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Nonlinear Devices
The primary application of nonlinear materials is the generation of new frequencies
not available with existing laser sources. The variety of applications for nonlinear
optical devices is so large that I will touch only the highlights here.
Second harmonic generation (SHG) received early attention primarily because of
early theoretical understanding and its use for measuring and testing the nonlinear
properties of crystals. Efficient SHG has been demonstrated using a number of
materials and laser sources. In 1968 Geusic et al (39) obtained efficient doubling of
a continuous w ave (cw) Nd: YAG laser using the crystal Ba2NaNbs01S' That same
year, Dowley (40) reported efficient SHG of an argon ion laser operating at
0.5145 J.lm in ADP. Later Hagen et al (41) reported 70% doubling efficiency of a
high energy Nd: glass laser in KDP (potassium dihydrogen phosphate), and Chesler
et aJ (42) reported efficient SHG of a Q-switch Nd: YAG laser using LiI03. An
efficiently doubled Q-switched Nd : YAG laser is now available as a commercial laser
source (43). In addition, LiI03 has been used to efficiently double a Ruby laser (44).
Recently the 10.6 pm COz laser has been doubled in Tellurium with .5% efficiency
(45) and in a ternary semiconductor CdGeAsz with 15% efficiency (46).
Three frequenc y nonlinear interactions include sum generation, difference fre­
quency generation or mixing, and parametric generation and oscillation. An
interesting application of sum generation is infrared up-conversion and image up­
conversion. For example, Smith & Mahr (47) report achieving a detector noise
equivalent power of 10- 14 W at 3.5 pm by up-converting to 0.447 tim in LiNb03
using an argon ion laser pump source. This detection method is being used for
infrared astronomy. Numerous workers have efficiently up-converted 10.6 11m to the
visible range (48-52) for detection by a photomultiplier. An extension of single beam
-
up-conversion is image up-conversion (37, 53, 54). Resolution to 300 lines has been
achieved, but at a cost in up-conversion efficiency.
Combining two frequencies to generate the difference frequency by mixing was
first demonstrated by Wang & Racette (15). Zernike & Berman (55) used this
approach to generate tunable far infrared radiation. Recently a number of workers
have utilized mixing in proustite
(56, 57),
(59), and recently AgGaSez (60, 61)
to
CdSe
generate
(58), ZnGeP2 (52, 52a), AgGaS2
tunable coherent infrared output
from near infrared or visible sources.
Perhaps the most unique aspect of nonlinear interactions is the generation of
coherent continuously-tunable laser-like radiation by parametric oscillation in a
nonlinear crystal. Parametric oscillators were well known in the microwave region
(62, 63) prior to their demonstration in the optical range. To date, parametric
oscillators have been tuned across the visible and near infrared in KDP (29, 64,
65) and ADP (66) when pumped at the second harmonic and fourth harmonic of
the 1.06 pm Nd: YAG laser, and they have been tuned over the infrared range from
0.6 J.lm to 3.7 tim in LiNb03 (67-72). The above parametric oscillators were
pumped by Q-switched, high peak power,laser sources. Parametric oscillators have
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1 50
BYER
also been operated in a cw manner in Ba2NaNbsOls (73-76) and in LiNb03 (77,
78). However, the low gains inherent in cw pumping have held back research in
this area.
In 1969 Harris (79) reviewed the theory and devices aspects of parametric
oscillators. Up to that time oscillation had been achieved in only three materials:
KDP, LiNb03, and Ba2NaNbsOls• Since 1969 parametric oscillation has been
extended to four new materials: ADP (80), Lil03 (81, 82), proustite (Ag3AsS3) (227),
and CdSe (83). The new materials have extended the available tuning range. However,
the development of oscillator devices still has remained materials limited. At this time
LiNb03 is the only nonlinear crystal used in a commercially available parametric
oscillator.! Smith (£4) and recently Byer (85) have discussed parametric oscillators
inreview papers and Byer (86) has reviewed their application to infrared spectroscopy.
Nonlinear interactions allow the extension of coherent radiation by second
harmonic generation, sum generation, and differew;e frequency mixing over a wave­
length range from 2200 A to beyond 1 mm in the far infrared. In addition, tunable
coherent radiation can be efficiently generated from a fixed frequency pump laser
source by parametric oscillation. The very wide spectral range and efficiency of
nonlinear interactions assures that they will become increasingly important as
coherent sources.
NONLINEAR PHENOMENA
Introduction
When a medium is suhjected to an electric field the electrons in the medium are
polarized. For weak electric fields the polarization is linearly proportional to the
applied field
P
=
toX1E
!
X
where
is the linear optical susceptibility and BO is the permittivity of free space
with the value 8.85 x 10- 1 2 F/m in mks units. The linear susceptibility is related to
the medium's index of refraction n by X!
n1
1.
In a crystaIIine medium the linear susceptibility is a tensor that obeys the
symmetry properties of the crystal. Thus for isotropic media there is only one value
of the index, and for uniaxial crystals two values, no the ordinary and ne the
extraordinary indices of refraction, and for biaxial crystals three values n" np, and
=
-
ny.
A linear polarizability is an approximation to the complete constitutive relation
which can be written as an expansion in powers of the applied field, as
P
=
co[Xl +X2. E+X3. E2+ . ]E
. .
where X2 is the second order nonlinear susceptibility and X3 is the third order
nonlinear susceptibility. A number of interesting optical phenomena arise from the
2
second and third order susceptibilities. For example, X gives rise to second harmonic
1 Chromatix Inc., Mountain View, California.
NONLINEAR OPTICAL PHENOMENA AND MATERIALS
151
generation (1), de rectification (87), the linear electro-optic effect or Pockels effect
(25), parametric oscillation (20), and three-frequency processes such as mixing (15)
and sum generation. The third order susceptib ility is responsible for third harmonic
generation (88), the quadratic electro-optic effect or Kerr effect (28), two-photon
absorption (89), and Raman (90), Brillouin (91), and Rayleigh (92) scattering.
We are primarily interested in effects that arise from X2• For a review of the
nonlinear susceptibility X2 and the resulting interactions in a nonlinear medium see
Wempl e & DiDomenico (93) and Ducuing & Flytzanis (93a).
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To see how XZ gives rise to second harmonic generation and other nonlinear
effects, consider an applied field
E
=
El cos(k1x -wt)+E2 cos(k2x-wt)
incident on the nonlinear medium. The nonlinear polarization is proportional to
XZEz, giving
tx2Ei[1 +cos(2klx-2wlt)]
plus a similar term for frequency (Oz. This term describes both dc rectification and
second harmonic generation. In addition, there are sum and mixing terms of the form
XZEIE2[cos{(kl-k2)X-(Wl-W2)t} +COS{(kl +k2)x-(W1 +W2)t}]
present in the expansion. These terms describe difference frequency and sum
frequency generation. All of the above processes take place simultaneously in the
nonlinear medium. The question that naturally occurs is how one process is singled
out to proceed efficiently relative to the competing processes. In nonlinear inter­
actions phasematching selects the process of interest to the exclusion of the other
possible processes. Thus, if the crystal birefringence is adjusted (by temperature or
angle of propagation) such that second harmonic generation is the phasematched
process, then it proceeds with relatively high efficiency compared to the remaining
processes invol ving sum and difference freq uency generation.
CRYSTAL SYMMETRY
Like the linear susceptibility, the second order nonlinear
susceptibility must display the symmetry properties of the crystal medium. An
immediate consequence of this fact is that in centrosymmetric media the second
order nonlinear coefficients must vanish. Thus nonlinear optical effects ar c restricted
to acentric materials. This is the same symmetry requirement for the piezoelectric
it tensors (94) and therefore the nonzero components of the second order
susceptibility can be found by reference to the listed it tensors. However, the
nonlinear coefficient tensors have been listed in a number of references (24, 35, 37,
95).
The tensor property of XZ can be displayed b y writing the nonlinear polarization
in the form
Pi(W3)
where Xijk(
=
-
So 2: Xijk: Ei(2)Ek(W1)
jk
(03. (Oz. (0 1) is the nonlinear susceptibility tensor.
1.
152
BYER
In addition to crystal symmetry restrictions, Xijk satisfies two additional symmetry
relations. The first is an intrinsic symmetry relation which can be derived for a
lossless medium from general energy considerations (23, 96). This relation states
that Xijk( ill3, ill2, ill!) is invariant under any permutation of the three pairs of indices
( ill3, i); (ill2,j); (ill!, k) as was first shown by Armstrong et al (19). The second
symmetry relation is based on a conjecture by Kleinman (2) that in a lossless medium
the permutation of the frequencies is irrelevant and therefore Xijk is symmetri� under
any permutation of its indices.
Finally, it is customary to use reduced notation and to write the nonlinear
susceptibility in terms of a nonlinear coefficient d;jk dim where m runs from 1-6
with the correspondence
-
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-
=
(11) (22) (33) (23) (13) (12)
m= 1
6
3
5
2
4
(jk)
=
and
Pi(W3)
=
Go
6
2.
I 2dim(EE)m
m=l
The 3 x 6 dim matrix operates on the column vector (EE)m given by
(EE)1
(EE)4
=
=
E;; (EEh E;; (EEh E;;
2Ey Ez; (EEh= 2Ex Ez; (EE)6
=
=
=
2Ex Ey
As an example, the nonlinear It tensor for the 42m point group to which KDP
and the chalcopyrite semiconductor crystals belong has the components
3.
However, Kleinman's symmetry conjecture states that d14 dl23 equals d36
since any permutation of indices is allowed. This is experimentally verified.
Equation 1 and 2 show that
=
d3 2
1
=
4.
This defines the relation between the nonlinear susceptibility and the It coefficient
used to describe second harmonic generation. The definition of the nonlinear
susceptibility has been discussed in detail by Boyd & Kleinman (97) and by
Bechmann & Kurtz (95).
'
MILLER S RULE
We have not yet made an estimate of the magnitude of the nonlinear
susceptibility. An important step in estimating the magnitude of a was taken by
Miller (21) when he proposed that the field could be written in terms of the
polarization as
E( -(3)
=
1
I2L\;jk( -W3' WI' Wz)PiWl.)Pk(WZ)
Go jk
-
Comparing Equations 5 and
dijk
=
2
shows that the tensor a and Ll are related by
So I
m Xil(W3)xjm(m2)XkM(ml)!�lmn( -(/)3 , Wz, (/) 1 )
I n
5.
6.
NONLINEAR OPTICAL PHENOMENA AND MATERIALS
153
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where Xij = (n�- 1) relates the linear susceptibility to the index of refraction. Miller
noted that � is remarkably constant for nonlinear materials even though iI varies
over four orders of magnitude.
Some insight into the physical significance of � can be gained by considering a
simple anharmonic oscillator model representation of a crystal similar to the Drude­
Lorentz model for valence electrons. This model has been previously discussed by
Lax et al (98), Bloembergen (23),Garrett & Robinson (99), and Kurtz & Robinson
(100). For simplicity we neglect the tensor character of the nonlinear effect and
consider a scalar model. The anharmonic oscillator satisfies an equation
x+rx+w�x+Ctx2
=
e
m
-
E(w, t)
where r is a damping constant, w6 is the resonant frequency in the harmonic
approximation, and a is the anharmonic force constant. Here E(w, t) is considered
to be the local field in the medium. The linear approximation to the above equation
has the well known solution
X(w)
where w;
=
n2-1
=
w;/(w�-w2-irw)
Ne2/mso is the plasma frequency. Substituting the linear solution back
into the anharmonic oscillator equation and solving for the nonlinear coefficient d
in terms of the linear susceptibilities gives
=
Ne3a
1
- fiom2 D(Wl)D(W2)D(W )
3
d-
where D(w) is the resonant denominator term in the linear susceptibility. Finally,
using the relation between � and d gi�en by Equation 6 we find that
�
=
(meorx)
7.
N2e3
On physical grounds we expect that the linear and nonlinear restoring forces are
roughly equal when the displacement x is on the order of the internuclear distance
a, or when w� u � au2• In addition, if we make the approximation that Na3 � 1 the
expression for � simplifies to
8.
For a = 2 A the value for Miller's delta predicted by our simple model is 0.25 m 2/C.
This compares very well with the mean value of 0.45 ± 0.07 m 2/e given by Bechmann
& Kurtz (95).
Equation 6 shows that the second order susceptibility to a good approximation
is given by
d
=
�
F.OX((J}3)x((J)2)X({J)1),1,
so(n2- 1)3,1,
�
son6,1,
In nonlinear processes d2/n3 is the material nonlinear figure of merit. Figure 1 shows
this figure of merit and the transparency range for a number of nonlinear materials.
154
BYER
IO.OOO �------" ------�
------ Te
�x� CdGeAs2
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1000
______
n
____
GaAs
-------.---- ZnGeP2
100
------- .---- AgGaSe2
TI3AsSe3
•
'"
Q
-- dSe
---------C
"
..,
c
"'.....
�
______ ______
10
AgGaS2
------ ----- Ag3AsS3
l-
ii
....
:::li:
.------ LiNb03
IJ..
0
W
II:
:;)
C>
·------ Li 103
lL
0.10
--________
ADP
. ------ Si0 2
O. OI L_
�__L___L___ L______L___L_ __�____L_�
____
30
20
10
5
2"
TRANSFr<RENCY RANGE
Figure 1
0.5
0.2
(I'm)
Figure of merit and transparency range: for selected nonlinear crystals.
NONLINEAR OPTICAL PHENOMENA AND MATERIALS
155
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Although d2/n3 varies over four- orders of magnitude, the intuitive physical picture
inherent in the anharmonic oscillator model gives a remarkably accurate account
of the magnitude of a material's nonlinear response.
THEORETICAL MODELS Although the anharmonic oscillator model gives insight into
the origin of the nonlinear susceptibility, it does not account for the tensor character
of a or allow the calculation of the nonlinear force constant rY..
The theory of the nonlinear susceptibility has been the subject of increased
consideration. Historically,a quantum treatment based on a perturbation expansion
of the susceptibility was the first description put forth for the nonlinear susceptibility
(19,23,24,101-104). Unfortunately,in solids, approximations to the quantum result
are required to obtain numerical results. In gases, where the wavefunctions are
better known, the quantum expressions for the third order susceptibility (second
order processes are not allowed by symmetry) do allow predictions of the magnitude
of X3 which agree very well with measured values (105).
Robinson (106) was the first to attempt to simplify the complete quantum
expression for the nonlinear susceptibility. He approached the problem by letting
the octapole moment of the ground state charge density serve as the eccentricity
of the electronic cloud. Flytzanis & Ducuing (107, 108) and Jha & Bloembergen
(109) applied this approach to the III-V compounds.
In the late 1960s Phillips & Van Vechten developed a theory for the dielectric
properties of tetrahedrally coordinated compounds (110, 111). This theory, based on
Penn's (112) earlier theory of the dielectric susceptibility, is the basis for several
models describing the nonlinear susceptibility of crystals (107, 110, 113-119).
Briefly, the theory is based on a single energy gap description of the material
which is related to the dielectric constant by
6(0)-1
=
Q�/E�
where Q; = hNe2/meo is the plasma energy of the valence (ground state) electrons.
Phillips & Van Vechten (110) have shown that the average energy gap Eg can be
decomposed into a covalent and ionic part Eh and C by the relation E; = E� + C2•
and that the ionicity of the bond is described by /; = C2/E;. For the ionic or
antisymmetric part of the bond, Phillips & Van Vechten (110) have shown that C
is described well by
where b is a constant, ZA and ZB are the valence of elements A and B which form
the bond, and ks is the Thomas-Fermi screening constant.
There are two approaches to calculating the nonlinear susceptibility based on the
above theory. The first is to interpret the theory as a two-band model and evaluate
the matrix elements appearing in the expansion for the second order susceptibility.
This approach has been taken by Flytzanis (107), Phillips & Van Vechten (110),
BYER
156
and Kleinman (116). Using the method, Kleinman shows that the nonlinear
coefficient for zincblende compounds is
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d14
=
( ) ( )�K�
a
0 .245 6 � i � �
lei ao Eh aoEg
9.
where ao is the Bohr radius and X is the linear susceptibility. The values of d14
predicted by Equation 9 are in good agreement wilth experiment.
The second approach (113, 114) calculates the nonlinear susceptibility from
variations in the linear susceptibility under an applied field. In this approach the
position of the bond charge can vary (the bond charge model) (113, 120) or a transfer
of charge along the bond axis can occur (the charge transfer model) (118, 119). Both
approaches give good descriptions of the second order susceptibility. The extension
to the third order nonlinear coefficient has been considered recently by Chemla et
al (121).
Levine (115) has modified and extended the bond charge model of Phillips &
Van Vechten (110). In summary, the bond charge theory does accurately account
for both the magnitude and the sign of the nonlinear susceptibility of most nonlinear
materials. However, for the purpose of searching for new nonlinear materials or
estimating the nonlinear coefficient of a potential nonlinear material, Miller's rule is
far simpler to apply.
Second H armonic Generation
We are now in a position to evaluate the nonlinear interaction in a crystal and to
calculate the conversion efficiency and its dependence on phasematching and
focusing. Conceptually, it is easier to treat the special case of second harmonic
generation and then to extend the principal results to three-frequency interactions.
The starting point for the analysis is Maxwell's equations from which the traveling
wave equation is derived in the usual manner. The traveling equation
V 2 E-/loaE-/l8E
•
••
••
=
/loP
1 0.
describes the electric field in the medium with a linear dielectric constant E driven
by the nonlinear driving polarization cPP/at2• It is customary to define the fields
by the Fourier relations
E(r, t)
=
�[E(r, w) exp{i(k' r-wt) } +c. c.]
and-
per, t)
=
�[P(r, w) exp{i(k' r- wt) } +c.c.]
Substituting into Equation 10 making the usual slowly varying amplitude approxi­
mations that alp p wi> p P, kDE/Dz P D2E/Dz2, and wE p E, and letting rx = tJ1uc
be the electric field loss per length, gives
oE
oz
-
+
IXE +
10E i}1ocwP
c at
2
-
- =
.:...__
.:e.
for the equation relating the envelope quantities of the fields. Neglecting loss,
NONLINEAR OPTICAL PHENOMENA AND MATERIALS
157
assuming a steady state solution, and using the definition given by Equation 3 for
the driving polarization, the above equation reduces to a pair of coupled nonlinear
equations for the fundamental and second harmonic waves
=
dE(w)
�
dE(2w)
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�
=
iKE(2w)E*(w)exp(i�kz)
lla.
iKE(w)E(w)exp(-i�kz)
lib.
where K = wdlnc and 11k = k(2w)-2k(w) is the wavevector mismatch. At phase­
matching 11k o and n(w) n(2w) since k 2nniA where n is the index of refraction.
Here we have introduced the interaction constant K which includes the nonlinear
coefficient d.
The above coupled equations for second harmonic generation have been
solved exactly (19). It is useful to discuss the solution for second harmonic generation
since the results can be extended to three-frequency interactions. We proceed by
considering the low conversion efficiency case for a nonzero 11k, and then the high
conversion efficiency case.
In the low conversion limit, the fundamental wave is constant with distance.
Therefore, we set dE(w)/dz = 0 and integrate Equation lib
=
=
f
E(z -I)
dE(2w)
=
o
fl/2
-1/2
=
iKE2(W)exp(-i�kz)dz
which gives
E(2w )1z=1
=
.
[exp(i�k[/2)- exp(i�k[/2)]
E2(w)
i�k
2 [sin(�kl/2)
KE (w)
(�kl/2)
II<
=
If we denote (sin x)/x by sinc x, note that the intensity is given by
I
=
ncco
2
IEI2
then in the low conversion limit, the conversion efficiency is
(
-
I(2w) r2[2 . 2 �kl
SlUe
I(w)
2
_
where
r2[2
=
KK IE(w)i2 [2
2w2Idl2 [2I(w)
3 3
n c eo
1 2.
13.
This example shows that phasematching enters into the nonlinear conversion
process through the phase synchronism factor sinc2 (l1klI2) which is unity at I1kl o.
=
1 58
BYER
Also, the second harmonic conversion efficiency is proportional to Idl2 and [2, as
expected, and varies as the fundamental intensity. The above result holds in the
plane wave focusing limit where 1= P/A and the area A TrW6/2 with Wo the
gaussian beam electric field radius.
For high conversion efficiencies, Equations l la and llb can be solved by
invoking energy conservation such that E2(W)+E2(2w) = Erne where Eine is the
incident electric field at z = O. The solution for perfect phasematching is the well
known result
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=
1(2w)
/(w)
=
tanh
2
(KEincZ)
which for small KEinc Z reduces to the low conversion efficiency result. In theory,
second harmonic generation should approach 100% as a tanh2 x function. In
practice, conversion efficiencies of 40-50% are reached under optimum focusing
conditions.
The solutions for second harmonic generation suggest that phasematching and
focusi ng are important if maximum conversion t:fficiency is to be achieved. The
important aspects of phasematching and focusing are discussed n ext.
PHASEMATCHING
It immediately follows from Equation 12 that for second har­
monic generation to be efficient, dk must be zero. As stated earlier, this is
accomplished in birefringent crystals by utilizing the birefringence to overcome the
crystal dispersion between the fundamental and second harmonic waves. As a
specific example, consider phasematching in the negative (ne < no) uniaxial crystal
KDP.
For a beam propagating at an an gle 8m to the crystal optic axis, the extraordinary
index is given by
[_1_ ]
ne(8m)
.2
cos2 8m
- n�
_
+
sin28m
n;
where no and ne are the ordinary and extraordinary indices of refraction at the
wavelength of interest. Two types of phasematching are possible for negative uniaxial
KDP:
Type I
Type II
For positive uniaxial crystals (ne > no) the extraordinary and ordinary indices are
reversed in the Type I and II phasematching expressions (35).
Type I phasematching uses the full birefringence to offset dispersion and Type
II phasematching averages the birefringence to use: effectively half of it in offsetting
the dispersion. A more important factor in choosing the type of phasematching
to be used is the nonlinear coefficient tensor. It must be evaluated to see that the
effective nonlinear coefficient remains nonzero. Thus for KDP where the 42m non­
linear tensor has components given by Equation 3, Type I phasematching requires
that the second harmonic wave be extraordinary or polarized along the crystallo­
graphic z axis. To generate this polarization, the fundamental waves must be
NONLINEAR OPTICAL PHENOMENA AND MATERIALS
159
polarized in the Ex and Ev direction and thus be ordinary waves. In addition, the
product Ex Ey should be maximized. Therefore, the propagation direction in the
crystal should be in the (110) plane at em to the crystal optic or z axis. For this
case, the effective nonlinear coefficient becomes deff d sin em and it is this coefficient
that is used in the SHG conversion efficiency expression. The effective nonlinear
coefficients for other crystal point groups have been calculated and listed (37, 97).
In addition, the phasematehing angles for a number of nonlinear crystals and
pump lasers have been listed by Kurtz (122).
The above discussion for uniaxial crystals can be extended to biaxial crystals.
As one might expect, the generalization becomes complicated. However, Hobden
(123) has given a complete description of the process including a careful definition
of the three principle indices na> np, and Hy
One other aspect of phasematl:hing is important in limiting SHG conversion
efficiency. As the extraordinary wave propagates in the crystal, its power flow
direction differs by the double refraction angle p from its phase velocity direction.
This walk-off of energy at the doubled refraction angle leads to a decrease in
SHG efficiency due to the separation of the ordinary fundamental wave and the
extraordinary second harmonic wave. The double refraction angle is given by
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=
p � tan p
= n;-2-(w) [n;1(2w) - n;1(2w])
.
sm
28m
Note that p = 0 at e 0 and e = n/2. The latter angle corresponds to a propagation
direction 90° to the crystal optic axis. Phasematching in this direction is referred to
a 90° phasematching and has the advantage of not inducing Poynting vector walk-off.
As shown in the next section, when p =F 0, the effective interaction length in the
crystal may be considerably reduced, thus reducing the conversion efficiency.
Therefore, 90° phasematching is desirable when possible.
=
FOCUSING
Up to this point we have assumed that the interacting waves are plane
waves of infinite extent. In fact, the beams are usually focused into the nonlinear
crystal to maximize the intensity and interaction length. In addition, the waves are
usually generated by laser sources and thus have a Gaussian amplitude profile with
electric field radius w. Gaussian beam propagation theory and laser resonators have
been treated by a number of authors and will not be considered here. For the
present discussion, the important factors of interest are (a) the form of the Gaussian
beam
E
=
Eo exp( r2/w2)
=
I· A
-
(b) the relation between the peak power and intensity
P
=
I
C;2)
and (c) twice the distance over which the beam area doubles
b
=
kw 2
=
2 nn 2
w
A
which is called the confocal distance.
14.
160
BYER
To include focusing, the SHG efficiency can be written in the form
I(2w)
I(w)
=
(
2wzd;ff
3
nnwnzwcoc
)
Pm Ikw h(B, !")
.,
15.
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where I is the crystal length, kw the wavevector in the crystal at the fundamental
frequency, and h(B, �) is the Boyd & Kleinman (97) focusing factor. The focusing
factor h(B, �) is a functioil of the double refraction parameter
and the focusing parameter
�
=
lib
where b is the confocal distance given by Equation 14.
In general, h(B,�) is an integral expression i nvolving B and �. However, it
reduces to simplified expressions in the proper limits. If we introduce the aperture
length la given by
w
1=
Jn
a
p
and an effective focal length
If -
-
nw2k - 3.b
2
-
If
given by
n
then the second harmonic conversion efficiency takes the limiting forms
[2
I(2w) KPw
I(w) - w2
_
lla
If la
41;
4.751}
for (la' If � 1)
for (If � f � fa)
for (l � If � la)
for (I � la � If)
for (Ta � 1 � if)
The first limit corresponds to the plane wave focusing case where h(�, B) � lib.
This case holds until lib"", 1, which is called the eonfocal focusing limit. The second
harmonic conversion efficient is optimum at lib 2.84 where h(2.84, 0) = 1.06 (97).
If P 0/= 0, then the aperture length may limit the interaction length to the second
and third cases. In practice, even a small walk-off angle may reduce the second
harmonic efficiency by 30 times for a given crystal length and focusing. It is there­
fore very desirable to adjust the crystal indices of refraction to achieve 90° phase­
matching if possible. In general, the last three of the above limits are not encountered
experimentally. This discussion on focusing is intended to serve as a general intro­
duction. A more detailed presentation is given by Boyd & Kleinman (97).
=
Three-Frequency Interactions
Three-frequency interactions include sum generation and difference frequency
generation or mixing in which two waves are incid,ent on the nonlinear crystal and
NONLINEAR OPTICAL PHENOMENA AND MATERIALS
161
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interact t o generate a third wave, and parametric generation in which a hIgh power
pump frequency interacts in a nonlinear crystal to generate two tunable frequencies.
We denote the three frequencies by WI' Wz, and W3 such that W3 = Wz +WI is
the energy conservation condition and k3 = kz +kl +�k is the momentum con­
servation condition with �k the momentum mismatch.
If we consider LiNb03 as an example, the generated polarization now has three
components given by
P(w1) = so2d31E(w3) E*(w 2)
P(w z) = Bo2d31 E(w3) E*(wl)
P(w3) = so2d31 E(w2)E(wl)
Substituting the polarization and electric field expressions at Wi' W 2 , and 0)3 into
the wave equation given by Equation 10 gives three coupled equations similar to
the two coupled equations previously derived for the SHG case :
d!l
+ (J(IEI
16a.
= K1E3E!exp(iilkz)
dEz
E = I.K2E3EI* exp( 'Ak )
dz + (J(z z
1D.
16b.
z
16c.
where E(w l ) is now written E 1 , Ki Wi dlni c, and !Xi is the loss.
We next investigate the solution of the above coupled equations for the three
frequency processes of interest. The coupled equations have been solved exactly
( 19, 1 24); however, we consider only the simplified case of interacting plane waves
and weak interactions such that the pump wave is not depleted.
=
(UP-CONVERSION) For the case of sum generation WI is a weak
infrared wave that sums with a strong pump wave at W2 to generate a high frequency
(visible) wave at W3. Negligible pump depletion implies that dE2/dz = 0 so that the
three coupled equations reduce to a pair of Equations (16a and 16c). If we assume
no loss, �k = 0, and a solution of the form efz with input boundary conditions that
E3(Z 0) = 0 and E I (z = 0) Et (0), the solution of the coupled equations is
SUM GENERATION
=
=
EI(z) = EI(O)COSrz
E3(z) = g EI(O) sin rz
\JW;
where
r
=
JKIK21E21z
Thus sum generation has an oscillatory solution with no net gain and a 100%
conversion for
rz
=
n12.
In
the low conversion limit, sin rz
conversion efficiency becomes
�
rz so
that the up­
1 62
3 (W3)'
WI
RVER
1
-- - 11(0)
_
smc
17.
2
where r2[2 (2Wl W3d2[2I )/(n1 n n3c3Bo) is equal to the previously derived SHG
2
2
conversion efficiency factor given by Equation 13. In the low conversion efficiency
limit the sum generation efficiency equals the SHG efficiency as one expects from
physical arguments. However, at high conversion efficiencies the sum generation
conversion oscillates while the SHG conversion approaches 100% as tanh2 rz.
Sum generation has been used to generate new frequencies in the same way as
SHG. For example, 10.6 /lm has been summed with 5.3 /lm in CdGeAs2 (125) to
generate the third harmonic of 10.6 11 at 3.5 11m. Similar summing has been used
to generate the third harmonic frequency of a 1.06 /lm Nd: YAG laser in KDP
with 60% efficiency (126).
Sum generation also allows infrared waves to be up-converted to the visible for
detection by a photomultiplier. Infrared up-conversion has been extensively studied
(127-129) and applied to astronomical uses (47).
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=
For difference frequency generation or mix­
ing the pump is the high frequency field at W3' Therefore, lack of pump depletion
implies dE3/dz = 0 so that the coupled equations r,�duce to Equations 16a and 16b.
Again assuming a solution of the form erz and boundary conditions that E1(z 0) ==
EI (0) and E (0) = 0 results in the solution
DIFFERENCE FREQUENCY GENERATION.
==
2
E1(z)
=
Ei(z)
=
,
E1(0) coshrz
� E1(O)sinhrz
.JW:
Now we notice that exponential functions have replaced the sin and cos functions
which occurred in the sum generation case. This implies that both the input field
at WI and the generated difference field at W grow during the nonlinear interaction
2
at the expense of the pump field. Again in the limit of low conversion efficiency
where sinh n "'" n the mixing efficiency becomes
�
11(0)
=
( )
(
Wz rZ[Zsincz Akl
2
WI
18.
3
where r2[2 = (2Wl w d2[2I3)/(nl n n3c 8o) is the conversion efficiency which equals
2
2
the SHG and sum generation efficiency. However, now both waves grow and have
net gain. This suggests the possibility of parametric oscillation once the gain
exceeds the losses.
Difference frequency generation or mixing has been used to measure parametric
gain (130) and to generate new frequencies. Of particular interest is the generation
of infrared and far infrared radiation by mixing in nonlinear crystals. A number of
experiments have recently been carried out using visible dye laser sources (56, 57)
and near infrared sources (59, 61) to generate tunable infrared output.
For this case we assume a strong pump wave at W3 and
equal input fields at Wl and w . Again the thn�e coupled equations reduce to
PARAMETRIC GENERATION
2
NONLINEAR OPTICAL PHENOMENA AND MATERIALS
163
Equations 16a and 16b in the absence of pump depletion. The remaining pair of
equations are the same as for mixing; however, now the boundary conditions that
E1(z = 0) = E1(0) and E2(z 0) E2(0) are inputs into the system. In fact, the input
fields need be only the quantum noise field associated with the gain of the parametric
amplifier.
The general solution of Equations 16a and 16b for the above boundary conditions
has been given by Harris (79) and by Byer (85) in their review articles on parametric
oscillators and is too long to reproduce here. However, for a single frequency
incident on a parametric amplifier, the gain defined by G2(l) (IE2(lJ!2/IE2(0J!2)-1
simplifies to
=
=
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=
C2(l)
=
inh2g l
gi
r212 s
where r2J2 = (2W1W2d2/213)/(nln2n3c3c;o) and 9
limit Equation 19 reduces to
G2 (
) = r212
l
19.
=
[r2-(�k/2nt. In the low gain
sinc2 (��l)
20.
which shows that the parametric gain equals the SHG conversion efficiency in this
limit. In the high gain limit Equation 19 reduces to
C2(l)
=
(1/4) exp(2rl)
2l.
for � k/2 � g. In practice high gains have been reported (65, 86, 131) such that super­
radiant operation is possible (126, 132).
Parametric amplifiers, like all linear amplifiers, have inherent noise. Since the
parametric frequencies can be in the visible range and are on the order of 1 0 - 10 W
per watt of pump power, the noise emission is intense enough to be easily visible
as parametric fluorescence (133, 134). A striking color photograph of parametric
fluorescence generated in LiNb03 (135) is reproduced in a review article by
Giordmaine (136). Parametric noise emission can be considered generated by the
amplified zero point fluctuations of the electromagnetic field (137-139). It can be
shown that the fluorescence noise power is given simply by Pnoise (energy per
22
photon) x gain x bandwidth or approximately by Pnoise
hw x r J x (c/2�nl) where
�n is the crystal birefringence and / the crystal length. For LiNb03 the emitted
noise power per watt is approximately 10-10 W, or one order of magnitude larger
than the spontaneous Raman scattering power. The noise power can be used to
measure the gain, bandwidth, and tuning characteristics of a parameter oscillator prior
to ever achieving threshold (133, 134). Parametric fluorescence also provides a very
accurate method of measuring the nonlinear coefficient of a crystal since only a power
ratio need be measured and not an absolute power as for SHG measurements (137).
However, one of the most interesting applications of parametric generation is the
achievement of coherent tunable laser-like output from an optical parametric
oscillator.
=
=
Parametric Oscillators
A parametric oscillator is schematically represented by a nonlinear crystal within
an optical cavity. The nonlinear crystal when pumped provides gain at the two
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1 64
BYER
frequencies Wz and WI (the signal and idler fields). When the gain exceeds the loss
the device reaches threshold and oscillates. At threshold the output power increases
dramatically, similar to the behavior of a laser. The generated output is coherent and
collinear with the pump laser beam. Once above threshold, the parametric oscillator
efficiently converts the pump radiation to continuously tunable signal and idler
frequencies.
There are a number of configurations for an optical parametric oscillator (OPO}.
The first distinction is between cw and pulsed operation. Due to the much higher
gains, we consider only pulsed operation where the pump is typically generated
by a Q-switched laser source. Parametric oscillators have also operated internal to
the pump laser cavity, but the external configuration is more common. Finally, there
are different parametric oscillator cavity configurations. The two most important
are the doubly resonant oscillator (DRO), where both the signal and idler waves
are resonated by the cavity mirrors, and the singly resonant oscillator (SRO), where
only one wave is resonant. The operation characteristics of parametric oscillators
with these cavity configurations have been discussed in detail in the review articles
by Harris (79), Smith (84), and Byer (85).
The important differences between DRO and SRO operation include threshold,
frequency stability, and pump laser frequency requirements. The threshold of a
DRO occurs when the gain equals the product of the signal and idler losses or
rZ[Z sincz
(Ll�l)
=
(Ll�l)
=
al az
22.
where rZf given by Equation 19 is the single pass parametric gain and az, a l are the
single pass power losses at the signal and idler. The SRO threshold is greater than
that of the DRO. If the idler wave at Wi is not resonated, then the SRO threshold
becomes
rZ[2 sincz
2a
z
23.
which is 2/a1 times greater than the DRO threshold. As an example consider a
5 em long 90° phasematched LiNb03 crystal pumped by the second harmonic of
a Nd : YAG laser at 0.532 /-lm. The DRO pump power threshold for 2% losses at
wt and Wz is only 38 mW. The S RO threshold is increased to 3.8 W. For cw pump
lasers the DRO is required in order to achieve threshold. However, for pulsed
parametric oscillator operation where kilowatts of pump power is available, the
higher SRO threshold is not a disadvantage and singly resonant offers significant
advantages over doubly resonant operation.
One obvious advantage of the SRO is thc much simpler mirror coating require­
ment since only one wave is resonated. The D RO also requires a single frequency
pump wave (79), whereas the S RO can be pumped by a multiple axial mode source
which more closely approximates typical laser characteristics. Finally, the DRO
operates with large frequency fluctuations since the condition wp = ws+wt must
hold for each parametric oscillator cavity mode and pump frequency. This leads to
mode jumping and the so-called "cluster effect" (67) in output frequencies where
165
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NONLINEAR OPTICAL PHENOMENA AND MATERIALS
only a cluster of a few adjacent axial modes oscillate within the gain lincwidth of
the parametric oscillator. Since the SRO has only a single cavity none of these
frequency competition effects occur. In fact, the SRO can be pumped with a multiple
axial mode pump laser source and still operate at a single frequency for the
resonated wave. These factors make singly resonant operation desirable whenever
possible.
The conversion efficiency of parametric oscillators has been studied in detail and
is discussed in the previously mentioned review papers. Briefly, the DRO theoretically
should be 50% efficient and the SRO near 100% efficient. In practice the DRO
efficiency approaches 50% and the SRO efficiency is near 40% although 60-70%
conversion efficiencies have been obtained.
The tuning and bandwidth of a parametric oscillator are determined by the
phasematching condition k3 kz + k\ and the sinc (�kl/2) phase synchronism factor.
=
1500
.,
E
E
u
c
1000 900
BOO
750
.... _ - ...
--- - - ..
---
20,000
LiNbO,
pump wavelengths.
z
...
d
>
�
650
600
550
500
TEMPERATURE
Figure 2
700
:J:
�
°c
450
parametric oscillator tuning curves for various doubled Nd : YAG laser
166
BYER
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In general, a parametric oscillator is tuned by altering the crystal birefringence. This
is usually done by crystal rotation, thus changing the extraordinary index of
refraction, or by controlling the crystal temperature and thus the birefringence. As
an example Figure 2 shows the temperature tuning curves of a LiNb03 parametric
oscillator pumped by various doubled wavelengths of a Q-switched Nd : VAG laser.
The tuning range extends from 0.55 to 3.7 J.lm. Tuning curves for a number of non­
linear crystals have been given in the review article by Byer (85).
The gain bandwidth of a parametric oscillator is found by analyzing the
2
sinc (�kl/2) function. Letting �k112 = n define the bandwidth gives
bw
=
2n/f31
where
is the crystal dispersion factor. Expanding f3 in terms of the crystal index of
refraction shows that f3 � 2�nlc so that the parametric oscillator gain bandwidth
in wavenumbers is approximately
I bv (em - 1)I
=
1
2�nl
24.
where �n is the birefringence and I the crystal length. A 5 cm long LiNb03
parametric oscillator has approximately a 1 cm - 1 gain bandwidth.
Like laser sources, the parametric oscillator can oscillate over a narrower
frequency range than the full gain bandwidth. In this regard, the parametric
oscillator acts like a homogeneously saturated laser and can operate in a single
axial mode without a reduction of the conversion efficiency. Linewidths as narrow
as O.(X)1 cm - l or 30 MHz have been achieved with LiNb03 parametric oscillators
(for further discussion see Byer, 85).
More than any other device the optical parametric oscillator requires a uniform
high optical quality, low loss nonlinear crystal. In addition, the crystal must have
adequate birefringence to phasematch and be able to withstand the high laser
intensities needed to reach threshold without damaging. These linear and nonlinear
optical material requirements are discussed in detail in the following section.
NONLINEAR MATERIALS
Material Requirements
Nonlinear crystals must satisfy four criteria if they are to be useful for nonlinear
optical applications. These criteria are adequate nonlinearity and optical trans­
parency, proper birefringence for phasematching, and sufficient resistance to optical
damage by intense optical irradiation. These properties are briefly discussed in the
next four sections. They are then illustrated by descriptions of useful nonlinear
crystals.
NONLINEARITY
In the early days of nonlinear optics adequate laser power was not
always available to take full advantage of the potential conversion efficiency of a
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NONLINEAR OPTICAL PHENOMENA AND MATERIALS
167
nonlinear crystal. Under those circumstances, the highest crystal nonlinearity was a
very important factor. Since the late 1960s the situation has changed chiefly due to
the availability of well controlled high peak power laser sources at many wave­
lengths across the ultraviolet, visible, and infrared spectral regions. Under present
circumstances, the crystal nonlinearity becomes only one factor in determining the
crystal's potential for nonlinear applications. For example, if more than adequate
laser power is available, then the nonlinear conversion efficiency is determined by
the maximum incident intensity the crystal can withstand without damaging. In
addition, the nonlinear conversion efficiency is then more usefully expressed at a
given intensity rather than power level. Finally, in comparing nonlinear crystals with
more than five orders of magnitude range in their figures of merit as illustrated
by Figure 1, one finds that most crystals have very nearly the same conversion
efficiency at a fixed intensity. This remarkable fact is illustrated by Figure 3 which
I L i N b03
1.0
(5 cm)
N
E
�
�
==
l-
0.10
«
I
I
C)
t;j
�
«
a::
«
0.01
�
CdGe S 2
<i
a::
C d Se
I
Z nGeP2 (I)
z
�
ZnGeP2 (n)
Te
I
ADP
(2 cm)
(5 cm)
AgGaSez
A gG S 2
a
TI3AsSe3
)..2
Ag 3 AsS 3
LiI03
Cl.
0.001
20
10
5
2
TRAN S PAR ENCY RANGE
Figure 3
0. 5
(JLm)
0. 2
Nonlinear conversion efficiency (parametric gain) at 1 MW/cm2 pump intensity
and transparency rahge of nonlinear crystals. The pump wavelength for parametric con­
version (second harmonic wavelength for SHG) is shown by the vertical tick mark. The
nonlinear efficiency varies as the square of the pump wavelength and crystal
crystal lengths are taken to be 1 cm unless specified.
length. The
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1 6g
BYER
shows the parametric gain for a number of nonlinear materials. Under the present
circumstances, the magnitude of a crystal's nonlinear coefficient is not of paramount
importance, but must be taken into account along with the other material parameters.
The measurement of crystal nonlinear coefficients has continued at a rapid pace
so that there are extended tables available listing nonlinear coefficients (35, 85, 95,
1 22). The measurement of absolute nonlinear coefficients has been carried out by a
number of methods including second harmonic ge:neration (140, 141), mixing (58),
and parametric fluorescence (137). The latter method has the advantage of being a
relative power measurement and not requiring the absolute measurement of power
as for SHG. Relative nonlinear coefficient measurements have also been made by
three methods : the powder technique (142), the Maker fringe method (4, 143), and
the wedge technique (144-146). The latter method is particularly useful in that it
gives, in addition to the relative magnitude of the nonlinear coefficient, the relative
sign and the coherence length of the interaction. At this time, absolute nonlinear
coefficients have been measured for a number of crystals including ADP, KDP,
LiI03, LiNb03, Ag3SbS3, CdSe, and AgGaSez. In the visible LiI03 and KDP and in
the infrared GaAs are used as standard nonlinear crystals. There is interest in
improving the accuracy of measurement for nonlinear coefficients and work is in
progress toward that goaL
Known nonlinear materials have transparency ranges that extend
from 2200 A in ADP through the infrared in a number of semiconductor compounds
and. beyond the reststrahl band into the far infrared in both oxide and semi­
conductor crystals. In general, the transparency range in a single material is limited
by the band edge absorption at high frequencies and by two-phonon absorption
at twice the reststrahlen frequency at low frequencil�s. Materials are also transparent
in the very low frequency range between de and the first crystal vibrational mode.
Thus nonlinear interactions may extend from the ultraviolet through the visible and
infrared to the far infrared. In fact, only a few I;rystals having the proper bire­
fringence and overlapping transparency ranges are needed to cover the entire
extended frequency range. This is important because the growth of high optical
quality nonlinear crystals is, in general, a difficult task. In addition, nonlinear
crystals that can be grown well, are transparent, and have proper birefringence for
phasematching are very rare among all known acmtric crystals (147). For example,
among 13,000 surveyed crystals only 684, or 5.25%, are uniaxial and phasematchable,
and of these less than half have a nonlinearity greater than KDP and far fewer are
amenable to crystal growth in high optical quality centimeter sizes.
The transmission loss in a nonlinear crystal reduces the SHG conversion
efficiency by e - (·2/ 2 +.1l1 where (i2 and (it are the loss per length at the second
harmonic and fundamental waves. Thus a 0.05 em " 1 and 0.025 em 1 loss at 2w and
w in a 1 em long crystal reduces the SHG efficiency by 0.95, which is negligible.
However, in a 5 em crystal the same losses reduce the efficiency by a factor of
0.77, which is significant. High optical quality oxide materials have losses in the
3
10 to 10- 5 em - 1 range, whereas semiconductor materials show much higher losses
in the 1 em - 1 to 10 - 2 em - 1 range. The reduction of optical loss in nonlinear
TRANSPARENCY
-
-
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NONLINEAR OPTICAL PHENOMENA AND MATERIALS
1 69
crystals is important if the nonlinear interaction is to proceed efficiently. It becomes
even more important for operation of a parametric oscillator where the threshold
pumping power and operating efficiency directly depend on the crystal and cavity
losses.
Recently intensity-dependent losses have been recognized as important factors in
the transparency of a crystal. At the short wavelength end of a material's trans­
parency range, two-photon absorption may become the dominant loss mechanism
even in a frequency range that is normally transparent to low intensity radiation.
For example, at 10 MW/cm2 the two-photon absorption in GaAs at 1 .32 !lm is
0.3 cm 1, which is significant compared to the 0.05 cm 1 normal absorption loss
( 148). This nonlinear absorption mechanism has also been observed in oxide non­
linear materials for intense pump radiation within the two-photon frequency range
of the band edge. Thus for maximum efficiency, pump wavelengths must be longer
than the two-photon absorption edge of the nonlinear crystal, although operation
near the band edge is possible at lower efficiencies and pump intensities. Gandrud
& Abrams (45) have reported an additional loss mechanism in tellurium due to
two-photon induced free carrier absorption. This mechanism reduced the SHG
conversion efficiency from 15 to 5 % for doubling a CO2 laser source.
The two-photon absorption limit was in mind when pump wavelengths for the
parametric gain calculations shown in Figure 3 were chosen. Figure 3 does illustrate
that with a few nonlinear materials, nonlinear interactions are possible over the
entire 2200 A to 30 !lm spectral region. Thus there is the possibility of generating
continuously tunable coherent radiation in nonlinear crystals over this extended
spectral range.
-
-
For efficient nonlinear interactions phase­
matching must be achieved in the nonlinear crystal. As an example, SHG in cubic
crystals that lack birefringence and thus are not phasematchable occurs only over
a distance of one coherence length between 10-100 !lm. By using birefringence to
offset dispersion, the phasematched interaction may proceed over the full crystal
length, or approximately 1 cm. Since the SHG efficiency varies as /2, phasematching
increases the efficiency by at least 104. The enormous improvement in nonlinear
conversion efficiency due to phasematching makes phasematching essential in non­
linear crystals. Thus if a crystal is nonlinear and transparent, it must still phasematch
to be useful. Adequate birefringence for phasematching is the most restrictive
requirement placed on a crystal and reduces the number of potential crystals to only
a few hundred out of over 1 3,000 known crystals.
In conducting surveys for new nonlinear crystals, the crystal symmetry is deter­
mined by X-ray methods or by reference to existing tables ( 149). If the crystal is
acentric and belongs to a point group that gives nonzero nonlinear coefficients for
phasematching (see Kurtz, 1 22, for further discussion) then the crystal indices of
refraction and birefringence must be determined. Fortunately, mineralogy texts and
data collections are available (150) which list indices of refraction and birefringence
for a large number of crystals. A comparison of the crystal birefringence against its
index of refraction and similar quantities for known phasematchable crystals gives
BIREFRINGENCE AND PHASEMATCHING
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1 70
BYER
a quick indication of whether the potential nonlinear crystal has adequate bire­
fringence for phasematching. Wemple (15 1), Wemple & DiDomenico (1 52), and Byer
( 147) have plotted birefringence vs index of refraction for known nonlinear crystals
to determine the minimum required birefringence for phasematching. Such plots are
useful since birefringence of a particular crystal cannot accurately be determined
theoretically at this time.
Of course the above preliminary indications of possible phasematching in a new
material must be carried a step further to determine the actual phasematching
properties of a crystal. This is usually done by a,xurately measuring the crystal
indices of refraction and fitting the results to an analytical expression which is useful
for phasematching calculatiO<ills. The measurement of crystal indices of refraction is
tedious experimentally, especially in the infrared, so that complete measurements
have been made on only a few crystals (35, 95). The measurements must be accurate
to 0.0 1 % for phasematching calculations to be made. For calculation purposes, the
index values vs wavelength are usually expressed by a Sellmeier equation or
modified Sellmeier equation. Once in this form, the phasematching expression for
SHG n� n�W(e) can be solved directly for the phasematching angle e. For three­
frequency phasematching conditions, a small computer program is usually written to
solve simultaneously the two equations 0) 3 Wz + w 1 and k3 = kz + k1 where k =
2nn(A)/A. Care must be taken in solving these equations since they may be double
valued. The LiNb03 parametric oscillator tuning curves, shown in Figure 2, were
calculated in this manner using analytical expressions for the index of refraction
given by Hobden & Warner ( 1 53).
Nonlinear crystals must also have a very uniform birefringence over the crystal
length for efficient nonlinear conversion. Birefringence uniformity places one of the
most stringent requirements on the growth of quality nonlinear crystals. Nash et
al (1 54) have discussed the reduction in conversion efficiency due to birefringent
nonuniformities.
Another limitation to nonlinear conversion efficiency is the breaking of phase­
matching due to thermally induced birefringence changes. This problem has been
treated in detail for SHG by Okada & Ieiri ( 155). They show that the optimum
phasematching temperature shifts with increasing average laser power. In crystals
with a large birefringence change with temperature, such as ADP and LiNb03,
thermal breaking of phasematching becomes serious at average power levels near
1 W. On the other hand, crystals such as LiI03 which have a small temperature
variation of birefringence can handle much higher average powers.
=
=
DAMAGE INTENSITY
An important limitation to the maximum nonlinear conversion
efficiency is crystal damage due to the input laser intensity. Laser induced damage
may be the result of a number of interactions in the material. Ready (1 56) discusses
possible damage mechanisms in a monograph on the effects of high-power laser
radiation. Among the mechanisms considered arc� thermal heating, induced ab­
sorption due to multiphoton absorption which leads to heating or to breakdown,
stimulated Brillouin scattering, self-focusing, surface preparation, and dielectric
breakdown. Thermal heating, in which the temperature rise of the material is
proportional to the input pulse duration or deposited energy is a common damage
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NONLINEAR OPTICAL PHENOMENA AND MATERIALS
171
mechanism i n metals and highly absorbing materials. Semiconductors behave more
like metals than dielectrics for this form of damage. Ready (1 56) lists semiconductor
damage thresholds (see p. 305 of ref. 1 56) as energy dependent near 5-10 Jjcm 2 . For
a 100 nsec laser pulse the damage intensity is thus 50-- 100 MW/cm 2 which agrees
with reported values for silicon and germanium (157). For 1 nsec pulses the power
density increases by 100 to greater than 1 GW/cm2 if thermal damage remains the
dominant mechanism. There is experimental evidence in GaAs that such high damage
thresholds are reached for pulses near 1 nsee. This has important implications for
infrared nonlinear optics since, in general, parametric gains are limited by the much
lower 20--60 MW/cm2 damage threshold for long pulses. In the limit of cw pumping,
semiconductor materials have damage thresholds as low as 1 kWjem2 due to surface
heating and melting.
Damage thresholds for dielectric materials are generally much higher than for
semiconductors. For example, sapphire damages near 25 GWjcm 2 for Q-switch pulse
irradiation. Glass & Guenther ( 1 58) have reviewed damage studies in dielectrics. They
point out that nonlinear materials show a marked decrease in damage threshold
for phasematched second harmonic generation. For example, LiI03 shows surface
damage at 400 MWjcm2 for 10 nsec pulses. When phasematched it damages at
30 MWjcm2 for the fundamental and 15 MWjcm2 for the second harmonic. Similar
results have been noted in LiNb03 and Ba2 NaNbs0 1 S '
In a series o f very careful studies, Ammann ( 1 59) measured LiNb03 surface
damage thresholds with various quarter wave coatings. He found that uncoated
LiNb03 damages at 40 MWjcm2 at 1 .08 /lm for a 0.5 W, 4 kHz, 140 nsec
repetitively Q-switch laser source. The damage threshold was increased to 1 50
MWjcm 2 for a )./4 layer of Al2 03-coated LiNb03 crystal. Additional measurements
on thin ZnS films showed the importance of overcoating and the lowering of the
damage threshold with time upon exposure to air ( 1 60). Similar lowering of damage
threshold for ThF 2 overcoated CdSe crystals used in an infrared parametric oscillator
has been noted (161). In that case the damage threshold decreased from 60 MW/cm 2
to 10 MWjcm 2 over a period of a few days. In addition, damage occurred much
more readily along surface scratches, indicating that surface preparation is important.
Bass & Barrett (162) proposed a probabilistic model for laser induced damage
based on an avalanche breakdown model. For this damage mechanism, the laser
field acts as an ac analog to dc dielectric breakdown. A laser power density of
10 GWjcm 2 corresponds to 4 x 106 Vjcm, which is close to the measured dc dielectric
breakdown fields near 30 x 106 Vjcm. Bass & Barrett ( 162) have presented the laser
damage threshold in a probabilistic way such that the probability to induce surface
damage is proportional to exp( - KjE) where K is a constant and E is the root
mean square (rms) optical field strength. Measured dielectric breakdown intensities
lie near 25 GWjcm 2 for glasses and fused silica and between 2--4 GWjcm 2 for non­
linear crystals. It appears that if other damage mechanisms to not limit the laser
intensity at lower levels, then laser induced dielectric breakdown determines the
maximum incident intensity. Efforts to understand the mechanisms of laser induced
damage have been increasing (163, 164). Hopefully these studies will lead to a better
understanding of the material's damage intensity limits.
Fortunately the damage intensity for most useful nonlinear materials lies between
1 72
BYER
10 MW/cm 2 and 1 OW/cm2. Figure 3 then shows that nonlinear conversion
efficiencies or parametric gains are high for crystal lengths on the order of 1 cm.
The damage intensity does place a limit on the · energy handling capabilities of
nonlinear crystals. Thus Q-switched laser pulse energies transmitted through 1 cm2
area crystals are limited to 1-10 J. For most applications of nonlinear crystals this
is not a serious limit since the laser host medium and the nonlinear material
damage at similar energy densities.
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KDP and
its Isomorphs
KDP, ADP, and their isomorphs are ferroelectric crystals ( 1 65) belonging to the
42m tetragonal point group above their Curie temperatures. KDP crystals are
transparent from 0.21 to 1 .4 11m for the nondeuterated and 0.21 to 1 .7 11m for the
deuterated material ( 166). ADP has a similar transparency range. KDP and its
isomorphs are negative birefringent and are phasematchable over most of their
t'ransparency range. For a complete reference to crystal indices of refraction see
Milek & Welles (166), Bechmann & Kurtz (95), and Singh (35).
Because of the availability of large, high optical quality crystals, KDP and ADP
were the subject of early nonlinear optical experiments by Miller et al ( 1 67) and
Miller (2 1). Later Francois (140) and Bjorkholm &. Siegman (141) made accurate
cw measurements of ADP's nonlinearity. In addition, Bjorkholm (168) has made
comparative nonlinear coefficient measurements of other crystals relative to KDP
and ADP.
KDP and ADP have played a significant role as efficient second harmonic
generators for both cw and pulsed sources. For cw doubling ADP and KDP can
be temperature tuned to the 90° phasematching condition over a limited range of
fundamental wavelengths between 0.54 pm and 0.49 pm at temperatures between
+ 60°C and - 80°C ( 1 69, 1 70). In particular, ADP and KDP 90° phasematch for
doubling 5 145 A at - 9.2°C and - 1 1.0°C. Other isomorphs of KDP 90° phasematch
over different wavelength regions. For example, Rubidium dihydrogen arsenate
(RDA) 90° phasematches for doubling the 0.694 11m Ruby laser (171) and Cesium
dihydrogen arsenate (CDA) 90° phasematches for doubling 1 .06 11m ( 1 72).
Using 90° phasematched ADP, Dowley & Hodges ( 1 69) obtained up to 100 mW
of 2573 A in 1 msec pulses and 30--50 mW of cw power. The doubling was
performed internal to an argon ion laser cavity to take advantage of the high
circulating fields. The SHO efficiency was strongly dependent on crystal losses.
The ultraviolet transparency and phasematching characteristics of KDP isomorph
crystals make them useful for ultraviolet generation by sum and second harmonic
generation. Huth et al (173) were the first to demonstrate this capability by
externally doubling a dye laser source. Similarly Yeung & Moore (174) and Sato
( 1 75) have generated tunable ultraviolet between 30144-3272 A by summing a Ruby
pumped dye laser and a Ruby laser.
Wallace (1 76) reported an intracavity doubled dye laser that tunes between 2610
and 3 1 50 A. This source uses a Q-switched internaHy doubled Nd : YAO laser as a
pump source for the rhodamine 60 and sodium fluorescein dye laser. The ADP
intracavity doubled dye laser produced 32 mW of average power at 2900 A in a
NONLINEAR OPTICAL PHENOMENA AND MATERIALS
1 73
em - 1 bandwidth. The conversion efficiency from input doubled Nd : YAG to
ultraviolet power was 4. 3 % . At the 65° phasematching angle used for doubling the
rhodamine 6G dye laser, the ADP double refraction angle is p = 0.025 rad, which
results in a conversion efficiency to the second harmonic of 1 % per 100 W of
input power.
Using a 1 J per pulse 10 pulse per second Nd : YAG laser source, Yarborough
( 1 77) has obtained high ultraviolet pulse energies and average powers. For the
1 .06 11m source Yarborough uses a flashlamp pumped 1/4" diameter rod Nd : Y AG
oscillator followed by 1/4" diameter and 3/8" diameter rod amplifiers. This source is
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2-3
doubled in a 90° phasematched CD* A crystal at 45% efficiency yielding 450 mJ,
0.532 !lm pulses. The CD*A phasematches at 103°C and the incident 1.06 !lm power
is unfocused into
the 20 mm CD* A crystal.
The green
0.532
11m
source is doubled
into the ultraviolet in a 2 cm long 90° phasematched ADP crystal. The conversion
efficiency for this step is 22%, yielding 100 mJ pulses at 0.266 !lm. The harmonic
generation in both CD*A and ADP takes place without damage to the crystals.
The power levels reported are limited by the laser source becoming unpolarized at
high pumping levels. Simultaneously, thermal heating of the doubling crystals begins
to limit the second harmonic generation efficiency. The ultraviolet source has been
used as a pump for an ADP parametric oscillator (66). In addition, the green
output at 0.532 11m has been used to pump dye lasers with up to 60% conversion
efficiency.
The high energy, high average power source reported by Yarborough illustrates
the optical quality of KDP and its isomorphs. In addition, the use of 90° phase­
matching for efficient second harmonic generation demonstrates its advantage in
these experiments. Although KDP-type crystals have not been utilized extensively
in parametric oscillator studies, they play an important role in generating tunable
ultraviolet radiation by second harmonic and sum generation of tunable visible
sources.
liNbO 3 and LiIO 3
LiNb03 is a ferroelectric material
its melting point of
with a Curie temperature approximately
Since the recognition of the unique
electro-optical ( 1 79) and nonlinear optical ( 1 80) properties of LiNb03 in 1 964, it has
been extensively studied ( 1 8 1 ) . The growth and physical properties of LiNb03 have
been discussed in a series of papers by Nassau et al ( 1 82) and by Abrahams et al
( 1 83-1S5). The electro-optic coefficients for LiNb03 have been measured by a
number of workers ( 1 86-1 89), as have the indices of refraction (1 53, 190, 1 9 1 ) . A
particularly useful form for the refractive indices, including temperature dependence,
is given by Hobden & Warner ( 1 53).
Early work with LiNb03 showed two potentially troublesome optical properties :
optically induced inhomogeneities in the refractive index ( 1 1 , 1 92-196) and growth­
dependent birefringent variations ( 197-200). The optically induced index inhomo­
geneities were found to be self-annealing for crystal temperatures above approxi­
mately I SO°C for visible radiation and 100°C for near infrared radiation. This fact
led to attempts to grow "hot" LiNb03 by adjusting the composition ( 198) and by
40°C below
( 1 78)
1 253°C.
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1 74
BYER
doping with MgO to increase the birefringence to achieve phasematching for 1 .06 Ilm
at temperatures approaching 1 80°C. In addition, new but related crystals such as
BaNaNbs0 1 s (39, 201) were grown in an attempt to overcome the index damage
problem. Unfortunately, these attempts were not successful due to the poor crystal
quality that resulted from doping LiNb03 and to the striations inherent in
Ba2 NaNbs0 1 s crystals.
Although growth striations severely limit the size and quality of Ba 2 NaNbs0 1 s
crystals, they have been used to double cw Nd : YAG lasers (39, 202) with good
efficiency. The nonlinear coefficient, which is approximately three times that of
LiNb03, partially offsets the disadvantage of smaller crystals of less optical quality.
Serious attempts to grow striation-free Ba2 NaNbs0 1 s crystals have not been
successful so that the use of this material is not widespread.
The growth-dependent birefringent variations in LiNb'o3 were less difficult to
eliminate. The problem was solved by growth of LiNb03 from its congruent melting
composition near a lithium to niobium ratio of 0.48 mole % (203-205). The growth
of LiNb03 crystals from a congruent melt plus improved optical quality tests ( 199,
206) led to uniform high quality single crystals of over 5 cm in length.
Ferroelectric LiNb03 has a large variation of birefringence with temperature.
This allows SHG at 90° phasematching for fundamental wavelengths between 1 and
3.8 Ilm at temperatures between O°C and 550°C. Conversely, LiNb03 90° phase­
matches for parametric oscillation for a number of pump wavelengths and can be
temperature tuned over a broad spectral region. Figure 2 shows the parametric
oscillator tuning curves for various pump wavelengths of an internally doubled Q­
switched Nd : Y AG laser. These LiNb03 phasematching curves were calculated by
using the Hobden & Warner ( 1 53) index of refraction expressions. Additional
LiNb03 tuning curves, including angular dependence, are given by Harris (79, 207)
and Ammann et al (208).
The internally doubled Q-switched Nd : YAG pumped LiNb03 parametric
oscillator is the best developed parametric oscillator at this time. This system is
described by Wallace (209). The Nd : YAG laser pump source operates with an
internal Lil03 doubling crystal, acollsto-optic Q-s.witch, and two Brewster angle
prisms for wavelength selection. The combination of four doubled Nd : YAG pump
wavelengths and temperature tuning allows the oscillator to cover the 0.54 to
Table 1
LiNb03
parametric oscillator operating parameters
Average
Pulse
Laser Pump
Tuning Range
Peak Power
Power
Length
Wavelength
(11 m)
(W)
(mW)
(nsee)
(11 m)
5-1 0
70
0.532
0.975-2.50
250-350
30
80
0.659
0.725-0.975
250
20
200
0.562
0.623-0.760
250
50
1 50
0.532
0.546-0.593
300
3-5
200
0.473
3.50-2.50
80-150
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NONLINEAR OPTICAL PHENOMENA AND MATERIALS
175
3.65 /tm spectral region. Table 1 gives the oscillator spectral output wavelengths,
laser pump wavelengths, and output power levels. The values used in this table are
useful for comparative purposes since they reflect the characteristics of the particular
Q-switched Nd : YAG pump laser and are not LiNb03 oscillator limits.
Threshold for the LiNb03 SRO with a 5 cm length 90° phasematched crystal is
typically between 300 and 600 W. The peak power conversion efficiency is near
50% for an oscillator operating a few times above the threshold. Due to the finite
pump pulse width and oscillator build-up time, the energy conversion efficiency is
approximately 30%. However, much higher conversion efficiencies have been
reported.
LiNb03 is one of the highest quality nonlinear optical materials. Its use in
parametric oscillators to generate tunable radiation over the 0.6 to 3.5 /tm spectral
range is a demonstration of its importance as a nonlinear material.
In 1968 Kurtz & Perry (142) applied a technique based on the measurement of
SHG of powders to the search for nonlinear materials. That search led to the
evaluation of a-iodic acid (a-HI03) for phasematched second harmonic generation
by Kurtz et al (210). Measurement of the nonlinearity of IX-HI03 showed that it had
a nonlinear coefficient approximately equal to that of LiNb03. The favorable non­
linear properties of IX-HI03 led to consideration of other AI03 crystals (21 1). One
of the first crystals considered was lithium iodate, LiI03. Its optical and nonlinear
optical properties were studied by Nath & Haussuhl (212) and Nash et al (2 13). The
more favorable optical quality of LiI03 compared to a-HI03 has led to its use in
a number of nonlinear applications spanning its entire transparency range from
0.35 to 5.5 /tm.
LiI03 (point group 6) has a measured nonlinear coefficient slightly greater than
that of LiNb03 (168, 171,214, 215). Its indices of refraction and large birefringence
(8n 0. 1 5) have been measured as have phasematching angles for various pump
wavelengths. Lil03 phasematches at 52° for doubling 0.6943 /tm of a Ruby laser
(44, 168,216) and at 30° for doubling a 1.06 /tm Nd : YAG laser.
Lil03 has proved particularly useful as a high optical quality nonlinear crystal
for internal second harmonic generation of a Nd : YAG laser. For this application
the laser mirrors are highly reflecting in the infrared but are transparent at the
second harmonic. To obtain efficient doubling the laser operates Q-switched. In
this mode, the LiI03 acts as a nonlinear output coupler and efficiently doubles the
Nd : YAG. By operating with a prism in the laser cavity any one of 15 Nd : YAG
laser lines can be selected and efficiently doubled by rotating the LiI03 to the phase­
matching angle. In this way wavelengths at 0.473, 0.532, 0.579, and 0.659 Jim can be
generated. For example, peak powers of over 10 kW and average powers of greater
than 1 W have been obtained from the internally doubled Nd : YAG laser source (72).
In 1970 Campillo & Tang (217) studied spontaneous parametric scattering in
LiI03 and Dobrzhanskii et al (218) carried out similar measurement in a-HIOz'
Shortly afterward, Goldberg (81) constructed a LiI03 parametric oscillator pumped
with a Ruby laser and Izrailenko et al (82) demonstrated an oscillator using LiI03
and IX-HI03 pumped by a doubled Nd : Glass laser.
Campillo (219) also externally doubled the Lil03 oscillator using an 8 mm Lil03
=
1 76
BVER
L i I 03 DOUBLING ANG L E
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w 35
..J
C)
Z
ex
..J
�
(/) 30
>0::
u
..J
ex
z
0::
w
I-
�
CAM P I L LO
20
Figure 4
1 . 1 and
WAV E L E NGTH
H ERBST
25
0.8
vs
�
1.4
1.5
1.6
FUNDAMENTAL WAVELE NGTH (/Lm)
1 .7
1.8
Measured Lil03 phasematching angles vs fundamental wavelength. Data between
1 . 8 Ilm from Campillo (219) and data between 0.9 and 1 . 3 Ilm from Herbst ( 1 6 1 ) .
crystal cut at 21.4°. Figure 4 shows the phasematching angles obtained in that
experiment over a range of fundamental wavelengths between 1 . 1 and 1 . 8 11m. The
second harmonic output at 100 W peak power tuned between 0.560 and 0.9 1 5 11m.
Figure 4 also shows phasematching data obtained by Herbst (161) in an internally
doubled LiNb03 parametric oscillator and illustrates the phasematching properties
of Lil03 for SHG over a broad spectral region.
The infrared transmission of Lil03 allows interactions out to 5.5 11m. Parametric
oscillation is possible, but not useful for idler wavelengths this far in the infrared
due to low gain. However, Lil03 does phasematch for mixing. Meltzer & Goldberg
(220) have demonstrated mixing in Lil03 internal to a Ruby pumped dye laser.
Output powers of 1 00 W were generated over the 4.1 to 5.2 JIm region by mixing
the Ruby source with the wavelengths from a DTTC dye laser. The spectral width
of the dye laser was 6 A in its 0.802 to 0.835 11m region. Although Lil03 is
transparent and phasematchable in the near infrared spectral region, its low effective
nonlinearity due to the 19° phasematching angle reduces its conversion efficiency to
the point that other materials may prove more useful for infrared generation by
mixing.
Semiconductor Nonlinear Materials
LiNb03 and Lil03 parametric oscillators have been tuned to 3.7 and 4.5 11m in the
infrared. Infrared oscillation is limited to these wavelengths due to the onset of
crystal absorption. Semiconductor nonlinear materials as discussed in this section
means the capability of extended infrared tuning from the two-photon absorption
NONLINEAR OPTICAL PHENOMENA AND MATERIALS
limit to beyond 10 /1m. All materials that satisfy this extended infrared transparency
range are semiconductors with bandgaps between 2 and 0.5 eV.
Tellurium was an early crystal used for SHG in the infrared (221-223). However,
problems in crystal growth and handling have held back its widespread use.
Tellurium has the largest nonlinear coefficient dll 649 X 10- 1 2 mfV of any known
crystal, but its large birefringence of i1n - 1 .45 and indices of refraction no = 6.25,
ne = 4.80 make it difficult to use and force phasematching to very small angles. In
spite of these difficulties, tellurium has been used to double a 10.6 /1m CO 2 laser
with 5% efficiency (45). Higher efficiencies were not possible due to induced
absorption loss by free electron generation by the intense fundamental and second
harmonic waves (45). This problem is intensified in tellurium due to its 4.0 flm
bandgap which permits two-photon absorption, thus generating the free carriers
which lead to free carrier absorption.
Proustite (Ag3AsS3) is an extensively studied infrared nonlinear crystal first
considered by Hulme et al (224). Its large birefringence allows phasematched
nonlinear interactions over its entire 0.6 to 13 flm transparency range. Proustite
has been used to double a 10.6 /1m CO 2 laser (225) and to up convert 10.6 ilm into
the visible using a Ruby laser pump source (49). Its indi�es of refraction have been
measured by Hobden (226). Proustite's large birefringence (i1n = 0.22) allows off­
angle phasematching near (J = 20°. Propagation at such a phasematching angle
results in a reduced effective nonlinear coefficient deff = d3 1 sin e + d2 2 cos e that is
approximately twice d3 1 for LiNb03. In addition, the large birefringence results in
a large double refraction angle (p 0.08), giving a short effective interaction length.
In spite of these drawbacks and the low crystal burn density near 25 MW/cm 2,
proustite has operated as a parametric oscillator near 2. 1 /1m pumped by a 1 .06 /1m
Q-switched source (227, 228). Recently Hanna et al (229) extended the operation
of a proustite parametric oscillator to the singly resonant configuration. They were
able to angle tune the oscillator over an extended 1.22 to 8.5 /1m spectral range.
However, operation of the infrared oscillator in proustite is still marginal due to
crystal damage problems and mirror coating problems. These difficulties can be
bypassed by mixing in proustite using a near infrared source (56, 57).
In addition to proustite, two closely related crystals have been investigated for
infrared nonlinear optical applications. These materials are pyrargyrite (Ag3SbS3)
(222, 224, 230) and Tl3AsSe3 described by Feichtner & Roland (23 1). Pyrargyrite
has properties very similar to proustite. It belongs to the same point group (3m),
is transparent between 0.6 and 13 flm and has a large enough birefringence to
phasematch over its transparency region. The measured phasematching angle for
doubling 10.6 /1m is 29°. Its nonlinear coefficient is very close to that of proustite.
The material Tl3AsSe3 appears useful as an infrared nonlinear material. Its
transparency range extends from 1 .26 to 17 /1m and the crystal has been grown
in sizes up to 1 .2 cm diameter and 3 em in length. The crystal has large bire­
fringence similar to proustite and phasematches for doubling 10.6 I�m at 22°. The
measured nonlinear coefficient is 3.3 times that of proustite and pyrargyrite. In
addition, Feichtner & Roland measured the crystal burn density at 10.6 /1m to be
32 MW/cm 2 compared to 20 MW/cm 2 for proustite and 14 MW/cm2 for pyrargyrite.
=
=
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1 77
=
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1 78
BYER
Unlike proustite and its related crystals, cadmium selenide (CdSe) 90° phase­
matches over an extended infrared region. Wurtzite (6 mm) CdSe does not have
adequate birefringence for second harmonic generation ; however, it does phase­
match for mixing and off-degenerate parametric oscillator operation (79). In addition
to its high optical quality, CdSe has a nonlinear coefficient three times that of
LiNb03·
Herbst & Byer (58) per formed the first phasematched interaction in CdSe by
mixing 10.6 11m against t h e 1 .833 11m Nd : YAG line (232) to generate 2.2 11m. The
measured angle for the allowed Type II phasematching was 77°. The observed
mixing efficiency was 35% in a 0.6 cm crystal at 48 MW/cm2, which is less than the
measured crystal burn density of 60 M W/cm2 .
The extended 0.75 to 25 11m transparency range of CdSe makes it useful as a
tunable mixing source between 8 and 25 /lm. In a recent experiment using a
LiNb03 parametric oscillator source, Herbst & Byer (see 86) gener ated tunable
radiation between 10 and 13.5 /lm by mixing in CdSe. The measured peak output
power of 2 W agreed with the expected conversion efficiency.
In an extension of the mixing experiment in CclSe, Herbst & Byer (83) demon­
strated the first singly resonant infrared parametric oscillator in CdSe pumped by
the 1.833 /lm line of Nd : YAG. Davydov et al (233) also reported achieving singly
resonant oscillation in CdSe. The observed threshold for the Herbst-Byer oscillator
was 550 W and pump depletions of up to 40% were obtained. Figure 5 is a photo-
Figure 5
Photograph of a CdSe singly resonant parametric oscillator operating at 2.2 and
10 11m in the infrared. The pump radiation is incident from the right through a lens. The
2
em long CdSe crystal within the
5.7 11m
confocal cavity is rotated for tuning.
Annu. Rev. Mater. Sci. 1974.4:147-190. Downloaded from www.annualreviews.org
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NONLINEAR OPTICAL PHENOMENA AND MATERIALS
1 79
graph of the CdSe parametric oscillator showing the 2 cm CdSe crystal within the
5.7 cm confocal cavity. At this time CdSe crystals up to 3 cm in length are
commercially available with longer (74 mm) higher-quality crystals having been
reported (234). CdSe singly resonant oscillators should operate reliably without
crystal burning for 4 cm long crystals.
Considerable effort has gone into growing new nonlinear materials, especially
materials that are useful in the infrared. One group of materials with the chalco­
pyrite (CuFeS 2 , point group 42m) structure (235) has particularly useful nonlinear
properties.
The chalcopyrite crystals form two groups which are ternary analogs of the
II-VI and III-V binary semiconductors. The first group is the 1-III-VI2 compounds
of which AgGaS 2 and AgGaSe2 are examples. The second group is the II-IV-V2
compounds which include CdGeAs 2 and ZnGeP2 . Goryunova et al (236) were the
first to investigate the semiconducting and nonlinear optical properties of chalco­
pyrite crystals. In 1971 Chemla et al (237) reported on AgGaS 2 and later that year
Boyd et al (238) measured the properties of ZnGeP 2 and carried out a phasematched
up-conversion experiment of 10.6 pm (52). Byer et al (239) measured the properties
of CdGeAs 2 • The properties of these chalcopyrites and other potentially useful
crystals of the same class were extensively studied by Boyd et al ( 146, 240, 241) in a
series of papers. As a result of these and other measurements (125, 242) four crystals
belonging to the chalcopyrite semiconductors have been identified as phasematch­
able and potentially useful for infrared nonlinear optics. These crystals are
CdGeAsz, ZnGePz, AgGaSez, and AgGaSz.
AgGaSz is transparent between 0.6 and 13 pm. It is negative birefringent and
phasematchable over a wide range of the infrared (237, 240, 241). At this time
crystals up to 1 cm in size have been grown, but optical quality is not yet sufficient
for use in parametric oscillators. The material has been used for infrared generation
by mixing and for second harmonic generation (59).
AgGaSe2 is a close analogy to AgGaS 2 . However, its transparency range (0.7317 /lm) and phasematching curves are shifted to longer wavelengths. Figure 6 shows
the calculated AgGaSe2 tuning curves based on the index of refraction data of Boyd
et al (146). Crystals have been recently grown up to 4 cm in length with 0.02 cm � 1
loss at 10.6 /lm. Although AgGaSe 2 has not been used in a parametric oscillator,
efficient SHG of 10.6 /lm and 7-15 /lm parametric mixing have been demonstrated
(61). AgGaSe 2 single crystals grow readily. This, along with its improved optical
quality, high nonlinearity, and infrared phasematching, make it a very useful infrared
nonlinear material (60).
Unlike the negativc bircfringent AgGaS 2 and AgGaSe2 , ZnGeP2 and CdGeAs2
are positive birefringent. Thus the Type II phasematching (n;wp n;(8)ws + niw;)
is allowed at 90° where Type I phasematching has a zero effective nonlinear
coefficient. However, the effective nonlinearity is maximum for Type I phasematching
[n� wp = n:(8)ws + nr(8)wJ at e = 45°. Type II phasematching doubles the bire­
fringence needed to achieve phasematching since it averages the birefringence at the.
longer wavelengths. ZnGeP 2 does not have enough birefringence to Type II phase­
match for SHG. However, it does phasematch for Type I SHG. Complete Type I
=
180
BYER
20
A g GaSe2
TYPE
I PHASEMATCHING
r-__�----__----------------�
ABSORPTION EDGE
-
- -
-
- -
-
- -
-
- - - -
Annu. Rev. Mater. Sci. 1974.4:147-190. Downloaded from www.annualreviews.org
by Zhejiang University on 12/29/11. For personal use only.
1 0 6..--E
::i..
:I:
t;z
5
�
3
w
...J
w
2
-- A p = 1 .50f' m
- A P " 1 . 3 1 8fL m
A p = 1 .06fLm
90
80
70
60
50
40
PHASEMATC H I N G ANGLE ( deg)
Figure 6
Calculated parametric oscillator tuning curves for AgGaSe z for various pump
wavelengths showing the extended 2-18 11m phasematching range.
SHG phasematching curves for ZnGePz are given by Boyd et al ( 146). ZnGePz
has been used for infrared up-conversion (52).
CdGeAsz has the highest parametric figure of merit of any known crystal except
Tellurium. In addition, it is particularly useful for SHG of a 10.6 11m CO2 laser
or for parametric oscillation with a 5.3 )lm pump source (239). Recent work (46)
has demonstrated the potential of CdGeAsz by SHG of a COz laser with 15%
efficiency in a 9 mm length crystal. Phasematched third harmonic generation is
also possible in CdGeAsz (243) and has been extensively studied both theoretically
and experimentally (121). CdGeAs2 crystals have been grown up to 9 mm in length
with a loss near 0.2 cm - 1 at 10.6 11m. Crystal size and optical quality improvements
are required before the full potential of this material can be realized.
At this time the chalcopyrite crystals satisfy all of the requirements for nonlinear
materials except low absorption loss and crystal uniformity. The crystals cannot yet
be grown in adequate sizes of high optical quality. However, the very useful
nonlinear properties, especially in the infrared, lend considerable importance to
chalcopyrite crystal growth efforts.
NONLINEAR OPTICAL PHENOMENA AND MATERIALS
181
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by Zhejiang University on 12/29/11. For personal use only.
CONCLUSION
The properties o f nonlinear materials are better understood when discussed with
reference to nonlinear devices and the theory of nonlinear interactions. Therefore,
in the first half of this review, I discussed nonlinear phenomena including the
origin of the nonlinear susceptibility and the theoretical models which describe
crystal nonlinearity. Nonlinear interactions which were briefly reviewed included
second harmonic generation and the three-frequency processes of sum and difference
frequency generation and parametric generation and oscillation. The important
aspects of focusing and the concept of phasematching were introduced.
With the above discussion as a frame of reference, the nonlinear material
rcquircmcnts including crystal nonlinearity, transparcncy, birefringence, and damagc
intensity were discussed. These properties were illustrated by describing useful
nonlinear crystals. The properties of KDP and ADP, LiNb03, LiI03, and semi­
conductor materials including proustite, pyrargyrite, cadmium selenide, and four
chalcopyrite materials were summarized. For reference Table 2 lists the linear and
nonlinear properties of selected nonlinear materials.
The first column of Table 2 gives the presently accepted values for the crystal
nonlinear coefficients. These values are based upon Levine & Bethea's (244) best
value for GaAs. The parametric gain is proportional to the material parameters
d2/n� n3 where d is the effective nonlinear coefficient at the angle of phasematching
and no and n3 are the indices of refraction for the degenerate and pump waves.
Column 8 of Table 2 lists the material figure of merit which is also shown in
Figure 1 along with the crystal transparency range.
In many applications the pump laser has more than adequate power. In this case
the quantity of interest is the nonlinear conversion efficiency per input intensity and
the crystal burn intensity. The calculated conversion efficiency r2{2 given in Table 2
assumes a 1 cm length crystal unless stated otherwise. Figure 3 shows the conversion
efficiency at 1 MW/cm2 pump intensity for the crystals listed in Table 2. The four
chalcopyrite crystals and LiNb03 and ADP show good efficiencies. An important
limitation to the maximum parametric gain is the material damage. Table 2 lists
approximate values for damage in nonlinear crystals based on reported damage
intensity measurements. Since laser induced damage may be the result of a number
of interactions in the material, the reported values vary widely. However, the listed
values are useful for estimating the conversion efficiency of a nonlinear crystal as
limited by the onset of damage.
The properties of a large number of nonlinear materials have been measured and
useful nonlinear crystals for the ultraviolet, visible, and infrared regions have been
characterized. Although the nonlinear crystals listed in Table 2 allow nonlinear inter­
actions over the spectral range from 0.25 to 25 11m, there are large classes of
materials that have not been investigated. Among these are organic materials (245)
and crystals potentially useful in the ultraviolet at wavelengths less than 0.2 11m.
There remains considerable work yet to be done for the full theoretical under­
standing of nonlinear crystals and the growth and improvement of presently known
phasematchable crystals. However, progress since the first experiment of Franken
182
BYER
Table 2
Nonlinear coefficient, figure of merit, conversion efficiency, burn intensity,
Material
(point group
pump wavelength)
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by Zhejiang University on 12/29/11. For personal use only.
Te (32)
A.p = 5.3 Jlm
CdGeAs2 ('12m)
Ap = 5.3 Jlm
GaAs (43m)
ZnGeP2 ('12m)
Ap = 1 .8 3 Jlm
Tl,AsSe, (3m)
Ap
=
1.83 Jlm
AgGaSe2 ('12m)
Ap = 1 .83 Jlm
CdSe (6 mn)
Ap = 1 .8 3 Jlm
AgGaSe2 ('12m)
Ap = 0.946 Jlm
Ag,SbS, (3m)
Ap
=
1 .06 Jlm
Ag3AsS3 (3m)
d x 1012
(m(V)
(References)
n.
d l l = 649
(222)
6. 25
4.80
d'6 = 236
(241, 239)
d36 = 90.1
(244)
d 36 = 75
(238)
d+ = 40
(23 1 )
d'6 = 33
( 1 46)
d 31 = 1 9
(238, 58)
d'6 = 1 2.
(240)
d+
=
12
(222)
n.
3.51
3 . 59
3.30
ne - no
em
1 . 4:5
14°
-
0
delf x 1 0 1 2
0.10
638
(d co s2 em)
55°
0.02 1
(dsinIJ)
I 35°
0.021
212
(dsin 2e)
-
-
-
II
+ 0.086
p
f--- -
.-
II 90°
0.0
1 62°
0.0 1
193
d36
3. 1 1
3.15
+ 0.038
3.34
3.15
- 0. 1 8 2
2 .62
2.5 8
- 0.3 2
2.45
2.47
+ 0.0 1 9
2.42
2.36
- 0.0:54
2.86
2 .67
- 0. 1 9
-
-
d+
-0.223
30°
oms
d+
2.76
36°
0.055
62 . 2
(d sin 2e)
d+
1 55°
0.01
27
(dsine)
1 90°
0.0
d3 6
90°
0.0
d' l
1 64°
0.17
1 0. 8
(d sin e)
1 90°
0.0
d3 6
Ap = 1 .06 Jlm
d+
LiIO, (6)
d' l = 7.5
(214)
1 .85
1 .72
- 0. 1 3 5
23°
0.071
d' l = 6.25
( 1 3 7)
2 . 24
2. 1 6
- 0.081
90°
0.0
d3 1
1 . 53
1 .48
- 0.0458
90°
0.0
d'6
- 0.04 1 7
90°
0.0
d36
+ 0.0095
-
_..
Ap = 0.694 Jlm
LiNb03 (3m)
Ap = 0.532
ADP (42m)
= 0.266
Ap
KDP (42m)
A p = 0.266
Si02 (32)
= 1 1 .6
(23 1 )
d 3 6 = 0.57
( 1 40, 1 4 1 )
d36 = 0.50
( 1 43)
dll
= 0. 3 3
( 143)
2.54
1.51
1 .47
1 .55
1 .56
- --
-
3.04
(dsine)
dl l
NONLINEAR OPTICAL PHENOMENA AND MATERIALS
and transmission range for nonlinear crystals
delr/nt,n3
X 10'4
r'l'
(I W)
0.0 1 1
0.95 x 1 0 - 4
3634
861
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by Zhejiang University on 12/29/11. For personal use only.
0.34
1039
226
1 87
r'l'
I(p)elr (em)
Icoh
= 1 04 11m
1 = I em
0.18
3.8 x 1 0 - 4
0.033
4.6 x 1 0 - 4
0.040
-
-
7.1 x 10- 3
1 .8 x 1 0 - 3
0.05
51
0.019
3.7 x 1 0 - 5
0.022
42
0.71
63.4
1 = I em
24
1=
2 cm
5.52 x 1 0 - 4
0.02
1 .98 x 1 0 - 3
0 .07
1 . 3 x 10-
3
Transmission
Range (11m)
4(}-60
4-25
20-40
2.4- 1 7
60
0.9
}
1 .4
- 17
0.21
0.59
1 27
[burn
(I MW/cm') ( M W/em')
0.09
>4
0. 7- 1 2
32
1 .2-18
> 10
0.73- 1 7
60
0.75-25
1 2-2 5
0.60-1 3
2.3 x 1 0 - 4
0.0 1 3
1 = 1 em
2.3 x 10-3
0.Q45
7.6
-
8 . 2 x 10- 6
0.01
1 4-50
0.60-14
8.2
0.007
9.0 x l O - 6
0.01 1
1 2-40
0.60-13
1 .88
0.008
5.6 x lO - 6
5.5 x l O - 3
1 25
0.3 1-5.5
9.2
11
0. 1 4
3.88
1=
5 em
2 . 1 x 10- 2
1 .28
50-140
0.35-4.5
0. 100
1 = 5 em
2.9 x 1 0 - 3
0.131
> 1000
0.20- 1 . 1
0.079
1 = 5 em
2.30 x 10- 3
0.103
> 1000
0.22-1 . 1
-
-
> 1000
0. 1 8-3.5
0.029
lcoh
=
14 11
183
Annu. Rev. Mater. Sci. 1974.4:147-190. Downloaded from www.annualreviews.org
by Zhejiang University on 12/29/11. For personal use only.
184
DYER
et al ( 1) has been rapid. I would like to condud.: this review by suggesting what
lies ahead in tunable coherent sources based on nonlinear interactions in crystals.
The parametric oscillator is a uniq ue tunable coherent source because its gain
mechanism due to the crystal nonlinearity is independent of its tuning and band­
width, which depend on the crystal birefringence and dispersion. Where tunable
laser sources typically operate over a 10% bandwidth (a dye laser for example) the
parametrtc oscillator tunes over a greater than two to one range. As an example,
Figure 7 shows the tuning curve of a 1 .06 /lm pumped LiNb03 parametric oscillator
which angle tunes between 1.5 /lm and 4.0 /lm. This oscillator was recently
demonstrated (246) and is capable of rapid tuning and high output energies.
The LiNb03 parametric oscillator's basic tuning range can be extended toward
the infrared by mixing the signal and idler in AgGaSe2 to cover the 3-1 8 /lm range
and by mixing in CdSe to tune over the 10-30 /lm range. Second harmonic generation
in LiNb03 and in LiI03 extends the tuning to 0.3-1.5 /lm. Finally, sum generation
4 r---_,-----.---r--.-.----�----r_--_,.
OSCILLATOR
TUNING CURVE
:3
OPTIC AX IS
MIRROR
L'o
L i Nb03
I
J
FtEFLECT I O N
R A NGE
\
� __� __ � � �
T " 126°C
O �____
__
�
__
__
__
. ____
48"'
__
__
�
__
__
CRYSTAL ORIENTA"nO N - - 8
Tuning curve for a 1.06 11m Nd : YAG pumped LiNb03 parametric oscillator.
mirror reflectance range for singly resonant operation is indicated.
Figure 7
The
NONLINEAR OPTICAL PHENOMENA AND MATERIALS
1 0 L....--�
(
1 85
LiNb0 3
PARAMETRI C
Annu. Rev. Mater. Sci. 1974.4:147-190. Downloaded from www.annualreviews.org
by Zhejiang University on 12/29/11. For personal use only.
OSCILLATOR
�1 :
)
90
L i Nb03
SHG
70
60
80
50
PHASE MATC H I N G
Figure 8
MIRROR RANGE
40
30
20
ANGLE (deg)
Spectral range vs crystal phasematching angle for the 1 .06 11m Nd : YAG pumped
LiNb03 parametric oscillator and following nonlinear crystal generators. CdSe and
AgGaSe2 phasematch for infrared generation by mixing the LiNb03 oscillator's signal and
idler frequencies. LiNb03 and Lil03 phasematch for doubling the primary oscillator
frequency range into the visible and ultraviolet.
in ADP pha sematches for generatio n of 0.22--0.3 11m in the ultraviolet. Figure 8
illustrates the phasematching angles an d tuning ranges for these nonlinear inter­
actions. A detailed study (247) shows that for 10 mJ pump energies available from a
1 .06 11m Q-switched Nd : YAG laser source, the parametric oscillator and all following
nonlinear interactions are 1{}-30% efficient. This widely tunable, high energy device
should operate very much like a coherent spectrometer source. The spectrometer
concept illustrates the unique capabilities of nonlinear interactions for generati on
of coherent radiation over an extended spectral range. The efficiency and high
power capability of nonlinear interactions assure wider application of nonlinear
devices as future tunable coherent sources.
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Manuscript
ANNUAL
REVIEWS
Further
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CONTENTS
EXPERIMENTAL AND THEORETICAL METHODS
NONDESTRUCTIVE TESTING METHODS, R. W. McClung
ApPLICATION
OF
NUCLEAR
MAGNETIC
RESONANCE
TO
SOLIDS:
HIGH
21
RESOLUTION TECHNIQUES, Robert W. Vaughan
STRUCTURE
SOME DEFECT STRUCTURES IN CRYSTALLINE SOLIDS, B. G. Hyde, A. N. Bagshaw,
Annu. Rev. Mater. Sci. 1974.4:147-190. Downloaded from www.annualreviews.org
by Zhejiang University on 12/29/11. For personal use only.
.
Sten Andersson, and M. O'Keeffe
43
PREPARATION, PROCESSING, AND STRUCTURAL CHANGES
93
ION IMPLANTATION, G. Dearnaley
PROPERTIES, PHENOMENA
125
OPTICAL EMISSION FROM SEMICONDUCTORS, H. J. Queisser and U. Heim
NONLINEAR OPTICAL PHENOMENA AND MATERIALS, Robert L. Byer
THEORIES PERTAINING TO
THE SEMICONDUCTOR-METAL TRANSITION
147
IN
CRYSTALS, L. L. Van Zandt and J. M. Honig
191
TRIBOLOGY: THE FRICTION, LUBRICATION, AND WEAR OF MOVING PARTS,
221
R. J. Wakelin
LIGHT SCATTERING IN NONCRYSTALLINE SOLIDS AND LIQUID CRYSTALS,
W. H. Flygare and T. D. Gierke
255
DIELECTRIC PROPERTIES OF CRYSTALS OF ORDER-DISORDER TYPE P. da R.
,
Andrade and S. P. S. Porto
FAST ION CONDUCTION, W. van Gool
287
311
SPECIAL MATERIALS
ONE- AND Two-DIMENSIONAL MAGNETIC SYSTEMS, Daniel W. Hone and
Peter M. Richards
337
STRONG, HIGH-TEMPERATURE CERAMICS, F. F. Lange
365
FIRE RETARDANT POLYMERS, G. L. Nelson, P. L. Kinson, and C. B. Quinn
391
BIOMEDICAL MATERIALS IN SURGERY, Donald J. Lyman and William J.
Seare, Jr.
REPRINT INFORMATION
415
434
INDEXES
AUTHOR INDEX
435
SUBJECT INDEX
.449
CUMULATIVE INDEX OF CONTRIBUTING AUTHORS, VOLUMES 1--4
459
CUMULATIVE INDEX OF CHAPTER TITLES, VOLUMES 1-4
460